This phase II trial is studying how well umbilical cord blood transplant from a donor works in treating patients with hematological cancer. Giving chemotherapy and total-body irradiation (TBI) before a donor umbilical cord blood transplant helps stop the growth of cancer and abnormal cells and helps stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from an unrelated donor, that do not exactly match the patient's blood, are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. Sometimes the transplanted cells from a donor can make an immune response against the body's normal cells (called graft-versus-host disease). Giving cyclosporine and mycophenolate mofetil before and after transplant may stop this from happening.

Adequate hepatic function; patients with clinical or laboratory evidence of liver disease will be evaluated for the cause of liver disease, its clinical severity in terms of liver function, histology, and the degree of portal hypertension; patients with fulminant liver failure, cirrhosis with evidence of portal hypertension or bridging fibrosis, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, or correctable hepatic synthetic dysfunction evidenced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin > 3 mg/dL, and symptomatic biliary disease will be excluded

If recent mold infection, e.g., Aspergillus, must be cleared by infectious disease

Second hematopoietic cell transplant: Must be >= 3 months after prior myeloablative transplant

Acute myeloid leukemia and acute lymphoblastic leukemia: Must have < 5% morphologic marrow blasts in an evaluable marrow (> 25% of normal cellularity for age) collected less than one month prior to start of conditioning; patients persistently aplastic for greater than one month since completing last chemotherapy are also eligible with the approval of the principal investigator (PI) or designee

Chronic myelogenous leukemia: All types, except refractory blast crisis; chronic phase patients must have failed or been intolerant to Gleevec or other tyrosine kinase inhibitors; at time of transplant, patients must have < 5% blasts in an evaluable marrow (> 25% of normal cellularity for age) by morphology within the bone marrow

Myelodysplastic syndrome (MDS): Any subtype; morphologic blasts must be less than 5% in an evaluable marrow (> 25% of normal cellularity for age); if blasts are 5% or more, patient requires induction chemotherapy pre-transplant to reduce blast count to less than 5%; patients who have a hypocellular marrow in the absence of excess blasts that is related to the underlying disease or as a result of treatment for MDS may also be eligible with the approval of the PI or designee

Large-cell lymphoma and aggressive T-cell lymphoma: With chemotherapy sensitive disease that has failed autologous transplant or patients who are ineligible for an autologous transplant; chemotherapy sensitive disease is defined as >= 50% reduction in the size of the tumor with the chemotherapy regimen immediately preceding transplant

Chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL): Must be refractory to fludarabine (fludarabine phoshate) or fail to have a complete or partial response after therapy with a regimen containing fludarabine (or another nucleoside analog, e.g. cladribine [2-CDA], pentostatin) or experience disease relapse within 12 months after completing therapy with a regimen containing fludarabine (or another nucleoside analog)

Follicular lymphoma, marginal zone B-Cell lymphoma, lymphoplasmacytic lymphoma, mantle-cell lymphoma, and indolent T-cell lymphomas: Patients with bulky disease should be considered for debulking chemotherapy before transplant; patients with refractory disease are eligible, unless they have bulky disease and an estimated tumor doubling time of less than one month

Multiple myeloma: Must have received prior chemotherapy; consolidation of chemotherapy by autografting prior to nonmyeloablative hematopoietic cell transplant (HCT) is permitted

Myeloproliferative syndromes

DONOR: Unit selection is based on the cryopreserved total nucleated cell (TNC) dose and matching at human leukocyte antigen (HLA)-A, B antigen level and DRB1 allele level typing; while HLA-C antigen/allele level typing is not considered in the matching criteria, if available, may be used to optimize unit selection

DONOR: The patient and the cord blood unit(s) must be matched for at least 4 of 6 loci

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Please refer to this study by its ClinicalTrials.gov identifier: NCT00723099